Comment by cooljoseph 2 days ago
The Hilbert curve does contain every point in the unit square. It is a limit of curves, and so can contain points even not in the intermediate constructions. This is similar to how the limit of 1/x as x -> infinity can be 0, even though 1/x never equals 0.
This doesn't seem to fly with the inductive fact that 1/2 of a power of two is always one over a power of two no matter how many times you perform the iteration.
There are a countably infinite number of rationals between any two rationals, you can even keep splitting up those rational infinitesimal gaps into countably many rationals that are infinitesimal even relative to the earlier infinitesimals.
And you still only end up with a countably infinite set of expressible locations and not the real continuum.
Either x, y, or both are guaranteed to be a number of that form for all values on the curve.
Also, a curve which gets arbitrarily close to every point in the unit square actually touches every point in the unit square. This is because (by definition) a curve is a continuous map from a compact space (the unit interval) to a Hausdorff space (R^2), and so its image is compact, and thus closed. A closed set contains every point that it is arbitrarily close to.